To most existing non-equilibrium theories, the modeling of non-isothermal processes was a hard task. Intrinsic difficulties involved the non-equilibrium temperature, the coexistence of conserved energy and dissipative entropy, etc. In this paper, by taking the non-isothermal flow of nematic liquid crystals as a typical example, we illustrated that thermodynamically consistent models in either vectorial or tensorial forms could be constructed within the framework of Conservation-Dissipation Formalism (CDF). And the classical isothermal Ericksen-Leslie model and Qian-Sheng model were shown to be special cases of our new vectorial and tensorial models in the isothermal, incompressible and stationary limit. Most importantly, from above examples, it was learnt that mathematical modeling based on CDF could easily solve the issues relating with non-isothermal situations in a systematic way. The first and second laws of thermodynamics were satisfied simultaneously. The non-equilibrium temperature was defined self-consistently through the partial derivative of entropy function. Relaxation-type constitutive relations were constructed, which gave rise to the classical linear constitutive relations, like Newton's law and Fourier's law, in stationary limits. Therefore, CDF was expected to have a broad scope of applications in soft matter physics, especially under the complicated situations, such as non-isothermal, compressible and nanoscale systems.
We derive the "conservation law" for nematic polymers in tensorial form valid for quadrupolar orientational order in contradistinction to the conservation law in the case of polar orientational order. Due to microscopic differences in the coupling between the orientational field deformations and the density variations for polar and quadrupolar order, we find that respective order parameters satisfy fundamentally distinct constraints. Being necessarily scalar in its form, the tensorial conservation law is obtained straightforwardly from the gradients of the polymer nematic tensor field and conn…
We explore the implications of the conservation law(s) and the corresponding "continuity equation(s)", resulting from the coupling between the positional and the orientational order in main-chain polymer nematics, by showing that the vectorial and tensorial forms of these equations are in general not equivalent and can not be reduced to one another, but neither are they disjoint alternatives. We analyze the relation between them and elucidate the fundamental role that the chain backfolding plays in the determination of their relative strength and importance. Finally, we show that the correct p…
We present and analyze correlation functions of a main-chain polymer nematic in a continuum worm-like chain description for two types of constraints formalized by the tensorial and vectorial conservation laws, both originating in the microscopic chain integrity, i.e., the conectivity of the polymer chains. In particular, our aim is to identify the features of the correlation functions that are most susceptible to the differences between the two constraints. Besides the density and director autocorrelations in both the tensorial and vectorial cases, we calculate also the density -- director cor…
An extension to a high-order model for the direct isotropic/smectic-A liquid crystalline phase transition was derived to take into account thermal effects including anisotropic thermal diffusion and latent heat of phase-ordering. Multi-scale multi-transport simulations of the non-isothermal model were compared to isothermal simulation, showing that the presented model extension corrects the standard Landau-de Gennes prediction from constant growth to diffusion-limited growth, under shallow quench/undercooling conditions. Non-isothermal simulations, where meta-stable nematic pre-ordering preced…
We show that the elastic charge of colloids in a nematic liquid crystal can be generated by the vector of external torque. The torque components play the role of two component charge (dyad) and give rise to the Coulomb-like potential, while their conservation law plays the role similar to that of Gauss' theorem in the electrostatics. The theory is applied to the colloid-surface interaction. A wall with homeotropic or planar director is shown to induce a repulsive 1/r^4 force on the elastic dipole. The external torque, however, induces the elastic charge in this colloid and triggers switching t…
We present a novel framework for the study of disclinations in two-dimensional active nematic liquid crystals, and topological defects in general. The order tensor formalism is used to calculate exact multi-particle solutions of the linearized static equations inside a uniformly aligned state. Topological charge conservation requires a fixed difference between the number of half charges. Starting from a set of hydrodynamic equations, we derive a low-dimensional dynamical system for the parameters of the static solutions, which describes the motion of a half-disclination pair, or of several pai…
Linear polymers and other connected "line liquids" exhibit a coupling between density and equilibrium nematic order on the macroscopic level that gives rise to a Meyer-de Gennes vectorial conservation law. Nevertheless, isotropic linear polymer melts/solutions exhibit fluctuations of the density and of the nematic order that are not coupled by this vectorial constraint, just like for isotropic liquids composed of disconnected non-spherical particles. It takes the proper tensorial description of the nematic order in linear polymer liquids, leading to a tensorial conservation law connecting dens…
Topological defects that form on surfaces of ordered media, dubbed boojums, are ubiquitous in superfluids, liquid crystals (LCs), Langmuir monolayers, and Bose-Einstein condensates. They determine supercurrents in superfluids, impinge on electrooptical switching in polymer-dispersed LCs, and mediate chemical response at nematic-isotropic fluid interfaces, but the role of surface topology in the appearance, stability, and core structure of these defects remains poorly understood. Here, we demonstrate robust generation of boojums by controlling surface topology of colloidal particles that impose…
Using a combination of isothermal-isobaric Monte Carlo and microcanonical molecular dynamics we investigate the relation between structure and self-diffusion in various phases of a model liquid crystal using the Gay-Berne-Kihara potential. These molecules are confined to a mesoscopic slit-pore with atomically smooth substrate surfaces. As reported recently [see M. G. Mazza {\em et al.}, Phys. Rev. Lett. {\bf 105}, 227802 (2010)], a reentrant nematic (RN) phase may form at sufficiently high pressures/densities. This phase is characterized by a high degree of nematic order and a substantially en…
Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associate…
In this paper, we proved that by choosing the proper variational function and variables, the variational approach proposed by M. Doi in soft matter physics was equivalent to the Conservation-Dissipation Formalism. To illustrate the correspondence between these two theories, several novel examples in soft matter physics, including particle diffusion in dilute solutions, polymer phase separation dynamics and nematic liquid crystal flows, were carefully examined. Based on our work, a deep connection among the generalized Gibbs relation, the second law of thermodynamics and the variational princip…
Stable array of point defects was generated in nematic liquid crystal. Point defects with topological charge of +1 and -1 (hedgehogs) were generated in vertically aligned liquid crystal cell. The hedgehogs were arranged in square or hexagonal arrays, and the shape of defect, such as radial or circular hedgehogs, were under the control of delicately patterned electrodes. Base on the two-dimensional defect array discovered in experiments, three-dimensional crystals of defects were postulated. The flux of the liquid crystal director field was summarized for each unit cell. The analysis showed tha…
The classical Lorentz reciprocal theorem (LRT) was originally derived for slow viscous flows of incompressible Newtonian fluids under the isothermal condition. In the present work, we extend the LRT from simple to complex fluids with open or moving boundaries that maintain non-equilibrium stationary states. In complex fluids, the hydrodynamic flow is coupled with the evolution of internal degrees of freedom such as the solute concentration in two-phase binary fluids and the spin in micropolar fluids. The dynamics of complex fluids can be described by local conservation laws supplemented with l…
We present Monte Carlo simulations of liquid-crystalline material confined to a nanoscopic slit-pore. The simulations are carried out under isothermal conditions in a specialized isostress ensemble in which N fluid molecules are exposed to a compressional stress acting on the fluid in directions parallel with the substrate planes. Fluid-fluid and fluid-substrate interactions are modelled as in our previous work [M. Greschek et al., Soft Matter, 2010, DOI:10.1039/B924417D). We study several anchoring mechanisms at the solid substrate by introducing an anchoring function (0<g(u)<1) that depends …
We present explicit expressions of the helicity conservation in nematic liquid crystal flows, for both the Ericksen-Leslie and Landau-de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified vorticity involving both velocity and structure fields (e.g. director and alignment tensor). This equation for the modified vorticity shares many relevant properties with ideal fluid dynamics and it allows for vortex filament configurations as well as point vortices in 2D. We extend all these results to particles of arbitrary shape by conside…
We explore the structure of nuclei and topological defects in the first-order phase transition between the nematic (N) and isotropic (I) phases in lyotropic chromonic liquid crystals (LCLCs). The LCLCs are formed by self-assembled molecular aggregates of various lengths and show a broad biphasic region. The defects emerge as a result of two mechanisms. 1) Surface anisotropy mechanism that endows each N nucleus (tactoid) with topological defects thanks to preferential (tangential) orientation of the director at the closed I-N interface, and 2) Kibble mechanism with defects forming when differen…
